Suppose the lady has $X . Initially.

When she went to first church, her money gets doubled. So she has $2X ,of which she donated $100 .

So before going to 2nd church she has $(2x-100)

Similarlly, in 2nd church doubled money will be 2(2x-100), after donating $100, She left with $(4x-300)

In 3rd church doubled money will be 2(4x-300), after donating $100, remaining money is $(8x-700)

In 4th church again doubled money will be 2(8x-700),after donating $100, the money left is $(16x-1500)=0

hence, 16x=1500,so x=$93.75

So initially she has $93.75